The Finite Element Method for Engineers

by Huebner, Kenneth H.; Dewhirst, Donald L.; Smith, Douglas E.; Byrom, Ted G.

• ISBN13: 9780471370789
• ISBN10: 0471370789
• Edition: 4th
• Format: Hardcover
• Copyright: 2001-09-07
• Publisher: Wiley-Interscience

Summary

A useful balance of theory, applications, and real-world examples The Finite Element Method for Engineers, Fourth Edition presents a clear, easy-to-understand explanation of finite element fundamentals and enables readers to use the method in research and in solving practical, real-life problems. It develops the basic finite element method mathematical formulation, beginning with physical considerations, proceeding to the well-established variation approach, and placing a strong emphasis on the versatile method of weighted residuals, which has shown itself to be important in nonstructural applications. The authors demonstrate the tremendous power of the finite element method to solve problems that classical methods cannot handle, including elasticity problems, general field problems, heat transfer problems, and fluid mechanics problems. They supply practical information on boundary conditions and mesh generation, and they offer a fresh perspective on finite element analysis with an overview of the current state of finite element optimal design. Supplemented with numerous real-world problems and examples taken directly from the authors' experience in industry and research, The Finite Element Method for Engineers, Fourth Edition gives readers the real insight needed to apply the method to challenging problems and to reason out solutions that cannot be found in any textbook.

Author Biography

KENNETH H. HUEBNER, PhD, is retired from Ford Motor Company, where he was manager of the Computer-Aided Engineering Research Staff. He received his PhD from Purdue University in 1969.<BR>

Table of Contents

Preface	xvii
PART I
Meet the Finite Element Method	3	(14)
What Is the Finite Element Method?	3	(2)
How the Finite Element Method Works	5	(3)
A Brief History of the Method	8	(3)
Range of Applications	11	(2)
Commercial Finite Element Software	13	(1)
The Future of the Finite Element Method	14	(3)
References	15	(2)
The Direct Approach: A Physical Interpretation	17	(57)
Introduction	17	(1)
Defining Elements and Their Properties	18	(22)
Linear Spring Systems	18	(3)
Flow Systems	21	(4)
Simple Elements from Structural Mechanics	25	(12)
Coordinate Transformations	37	(3)
Assembling the Parts	40	(16)
Assembly Rules Derived from an Example	40	(6)
General Assembly Procedure	46	(1)
Features of the Assembled Matrix	47	(1)
Introducing Boundary Conditions	48	(8)
Solver Technology	56	(7)
Linear Direct Solvers	57	(1)
Iterative Solvers	58	(2)
Eigensolvers	60	(1)
Nonlinear Equation Solvers	60	(3)
Closure	63	(11)
References	63	(1)
Problems	64	(10)
The Mathematical Approach: A Variational Interpretation	74	(39)
Introduction	74	(1)
Continuum Problems	75	(4)
Introduction	75	(1)
Problem Statement	76	(2)
Classification of Differential Equations	78	(1)
Some Methods for Solving Continuum Problems	79	(6)
An Overview	79	(1)
The Variational Approach	80	(1)
The Ritz Method	81	(4)
The Finite Element Method	85	(23)
Relation to the Ritz Method	85	(1)
Generalizing the Definition of an Element	86	(1)
Example of a Piecewise Approximation	87	(4)
Element Equations from a Variational Principle	91	(2)
Requirements for Interpolation Functions	93	(6)
Domain Discretization	99	(3)
Example of a Complete Finite Element Solution	102	(6)
Closure	108	(5)
References	108	(1)
Problems	109	(4)
The Mathematical Approach: A Generalized Interpretation	113	(24)
Introduction	113	(1)
Deriving Finite Element Equations from the Method of Weighted Residuals	114	(17)
Example: One-Dimensional Poisson Equation	119	(6)
Example: Two-Dimensional Heat Conduction	125	(4)
Example: Time-Dependent Heat Conduction	129	(2)
Closure	131	(6)
References	131	(1)
Problems	132	(5)
Elements and Interpolation Functions	137	(86)
Introduction	138	(1)
Basic Element Shapes	139	(5)
Terminology and Preliminary Considerations	144	(2)
Types of Notes	144	(1)
Degrees of Freedom	144	(1)
Interpolation Functions-Polynomials	144	(2)
Generalized Coordinates and the Order of the Polynomial	146	(5)
Generalized Coordinates	146	(1)
Geometric Isotropy	147	(2)
Deriving Interpolation Functions	149	(2)
Natural Coordinates	151	(10)
Natural Coordinates in One Dimension	152	(1)
Natural Coordinates in Two Dimensions	153	(4)
Natural Coordinates in Three Dimensions	157	(4)
Interpolation Concepts in One Dimension	161	(5)
Lagrange Polynomials	161	(2)
Hermite Polynomials	163	(3)
Internal Nodes--Condensation/Substructuring	166	(4)
Two-Dimensional Elements	170	(14)
Elements for C0 Problems	170	(8)
Elements for C1 Problems	178	(6)
Three-Dimensional Elements	184	(5)
Elements for C0 Problems	184	(5)
Elements for C1 Problems	189	(1)
Isoparametric Elements for C0 Problems	189	(8)
Coordinate Transformation	190	(3)
Evaluation of Element Matrices	193	(3)
Example of Isoparametric Element Matrix Evaluation	196	(1)
Numerical Integration	197	(13)
Newton-Cotes	199	(3)
Gauss-Legendre	202	(3)
Numerical Example for Element Matrix Evaluation	205	(5)
Closure	210	(13)
References	211	(2)
Problems	213	(10)
PART II
Elasticity Problems	223	(65)
Introduction	224	(1)
General Formulation for Three-Dimensional Problems	224	(14)
Problem Statement	224	(2)
The Variational Method	226	(6)
The Galerkin Method	232	(4)
The System Equations	236	(2)
Application to Plane Stress and Plane Strain	238	(8)
Displacement Model for a Triangular Element	238	(2)
Element Stiffness Matrix for a Triangle	240	(3)
Element Force Vectors for a Triangle	243	(3)
Application to Axisymmetric Stress Analysis	246	(8)
Displacement Model for Triangular Toroid	247	(1)
Element Stiffness Matrix for Triangular Toroid	247	(3)
Element Force Vectors for Triangular Toroid	250	(4)
Application to Plate-Bending Problems	254	(8)
Requirements for the Displacement Interpolation Functions	257	(1)
Rectangular Plate-Bending Elements	258	(4)
Three-Dimensional Problems	262	(2)
Introduction	262	(1)
Formulation for the Linear Tetrahedral Element	262	(1)
Higher-Order Elements	263	(1)
Introduction to Structural Dynamics	264	(14)
Formulation of Equations	264	(3)
Free Undamped Vibrations	267	(2)
Finding Transient Motion via Mode Superposition	269	(3)
Finding Transient Motion via Recurrence Relations	272	(6)
Closure	278	(10)
References	279	(2)
Problems	281	(7)
General Field Problems	288	(60)
Introduction	289	(1)
Equilibrium Problems	289	(14)
Quasi-Harmonic Equations	289	(1)
Boundary Conditions	290	(3)
Variational Principle	293	(1)
Element Equations	294	(3)
Element Equations in Two Dimensions	297	(6)
Eigenvalue Problems	303	(8)
Helmholtz Equations	304	(1)
Variational Principle	305	(1)
Element Equations	306	(1)
Examples	307	(2)
Sample Problem	309	(2)
Propagation Problems	311	(9)
General Time-Dependent Field Problems	311	(5)
Finite Element Equations	316	(2)
Element Equations in One Space Dimension	318	(2)
Solving the Discretized Time-Dependent Equations	320	(15)
Solution Methods for First-Order Equations	321	(1)
Finding Transient Response via Mode Superposition	321	(3)
Finding Transient Response via Recurrence Relations	324	(3)
Oscillation and Stability of Transient Response	327	(3)
Algorithm Order	330	(2)
Sample Problem	332	(3)
Closure	335	(13)
References	335	(2)
Problems	337	(11)
Heat Transfer Problems	348	(74)
Introduction	349	(1)
Conduction	349	(30)
Problem Statement	349	(2)
Finite Element Formulation	351	(5)
Element Equations	356	(9)
Linear Steady-State and Transient Solutions	365	(6)
Nonlinear Steady-State Solutions	371	(4)
Nonlinear Transient Solutions	375	(4)
Conduction with Surface Radiation	379	(11)
Problem Statement	379	(2)
Element Equations with Radiation	381	(3)
Steady-State Solutions	384	(2)
Transient Solutions	386	(4)
Convective-Diffusion Equation	390	(8)
Problem Statement	391	(1)
Finite Element Formulation	391	(2)
One-Dimensional Problem	393	(4)
Two-Dimensional Solutions	397	(1)
Free and Forced Convection	398	(5)
Problem Statement	398	(1)
Finite Element Formulation	399	(2)
Solution Techniques	401	(1)
Free-Convection Example	402	(1)
Forced Convection	403	(1)
Closure	403	(19)
References	406	(4)
Problems	410	(12)
Fluid Mechanics Problems	422	(73)
Introduction	422	(1)
Inviscid Incompressible Flow	423	(11)
Problem Statement	424	(1)
Finite Element Formulation	425	(3)
Velocity Component Smoothing	428	(1)
Example with Unstructured Mesh	429	(4)
The Kutta Condition	433	(1)
Viscous Incompressible Flow without Inertia	434	(7)
Problem Statement	434	(2)
Stream Function Formulation	436	(1)
Velocity and Pressure Formulation	437	(4)
Viscous Incompressible Flow with Inertia	441	(18)
Mixed Velocity and Pressure Formulation	442	(6)
Penalty Function Formulation	448	(3)
Equal-Order Velocity and Pressure Formulation	451	(8)
Compressible Flow	459	(20)
Problem Statement	460	(1)
Low-Speed Flow with Variable Density	461	(6)
High-Speed Flow	467	(12)
Closure	479	(16)
References	480	(6)
Problems	486	(9)
Boundary Conditions, Mesh Generation, and Other Practical Considerations	495	(75)
Introduction	496	(1)
Physical Singularities	496	(2)
Benchmark Problems	498	(4)
Symmetry	502	(15)
Definition of Types of Symmetry	503	(7)
Antisymmetry	510	(2)
A Complex Loading Example: Two Axes of Symmetry	512	(1)
Axisymmetry and Rotational Symmetry	512	(1)
The Torsion Problem	513	(3)
Heat Transfer	516	(1)
Dimensional Analysis	517	(2)
Mesh Generation	519	(22)
Mapped Meshing	520	(1)
Free Meshing	521	(5)
Mesh Topology Cleanup and Mesh Smoothing	526	(3)
Mesh Refinement Methods	529	(2)
Error Indicators	531	(4)
Adaptive Remeshing	535	(2)
p and h/p Methods	537	(4)
Lumped Mass versus Consistent Mass	541	(2)
Modeling Fasteners	543	(3)
Connecting Rod Analysis	546	(3)
Crankshaft and Flywheel Analysis	549	(2)
Disc Brake Analysis	551	(7)
An Automotive Brake Primer	552	(1)
Rotor Analysis Using Rotational Symmetry	553	(3)
Rotor Coning Analysis Using Axisymmetry	556	(1)
Simulating Friction	557	(1)
Hot Spotting	558	(1)
Closure	558	(12)
References	559	(6)
Problems	565	(5)
Finite Elements in Design	570	(66)
Introduction	571	(1)
Design Optimization	571	(11)
The Optimization Problem	572	(2)
Practical Aspects of Numerical Optimization	574	(3)
Optimization Algorithms	577	(4)
Software Packages for Optimal Design	581	(1)
Finite Element-Based Optimal Design	582	(12)
Design Parameterization	583	(3)
Structural Optimization	586	(3)
Topology Optimization	589	(3)
Approximation Techniques	592	(1)
Multidisciplinary Design Optimization	593	(1)
Design Sensitivity Analysis	594	(12)
Finite Difference Approximations	594	(1)
Analytical Methods for Design Sensitivity Analysis	595	(5)
Design Sensitivities for Eigenproblems	600	(5)
Semianalytical Approach	605	(1)
Other Advancements in Design Sensitivity Analysis	606	(1)
Examples of Design Sensitivity Analysis	606	(20)
Numerical Example: Steady-State Equilibrium Analysis	606	(5)
Numerical Example: Eigenvalue and Eigenvector Analysis	611	(4)
Sensitivity Analysis for Steady-State Conduction in a Solid	615	(8)
Numerical Example for Steady-State Conduction in a Solid	623	(3)
Case Study: Finite Element-Based Design	626	(2)
Closure	628	(8)
References	628	(4)
Problems	632	(4)
Appendix A Matrices	636	(12)
A.1 Definitions	637	(1)
A.2 Special Types of Square Matrices	637	(1)
A.3 Matrix Operations	638	(2)
A.4 Special Matrix Products	640	(1)
A.4.1 Product of a Square Matrix and a Column Matrix	640	(1)
A.4.2 Product of a Row Matrix and a Square Matrix	640	(1)
A.4.3 Product of a Row Matrix and a Column Matrix	641	(1)
A.4.4 Product of the Identity Matrix and Any Other Matrix	641	(1)
A.5 Matrix Transpose	641	(1)
A.6 Quadratic Forms	641	(2)
A.7 Matrix Inverse	643	(1)
A.8 Matrix Partitioning	643	(2)
A.9 The Calculus of Matrices	645	(1)
A.9.1 Differentiation of a Matrix	645	(1)
A.9.2 Integration of a Matrix	645	(1)
A.9.3 Differentiation of a Quadratic Functional	645	(1)
A.10 Norms	646	(2)
Appendix B Variational Calculus	648	(9)
B.1 Introduction	648	(1)
B.2 Calculus--The Minima of a Function	648	(2)
B.2.1 Definitions	648	(1)
B.2.2 Functions of One Variable	649	(1)
B.2.3 Functions of Two or More Variables	650	(1)
B.3 Variational Calculus--The Minima of Functionals	650	(7)
B.3.1 Definitions	650	(1)
B.3.2 Functionals of One Variable	651	(4)
B.3.3 More General Functionals	655	(1)
References	656	(1)
Appendix C Basic Equations from Linear Elasticity Theory	657	(17)
C.1 Introduction	657	(1)
C.2 Stress Components	658	(1)
C.3 Strain Components	658	(1)
C.4 Generalized Hooke's Law (Constitutive Equations)	659	(3)
C.5 Static Equilibrium Equations	662	(1)
C.6 Compatibility Conditions	663	(1)
C.7 Differential Equations for Displacements	664	(1)
C.8 Minimum Potential Energy Principle	665	(2)
C.9 Plane Strain and Plane Stress	667	(3)
C.10 Thermal Effects	670	(1)
C.11 Thin-Plate Bending	671	(3)
References	673	(1)
Appendix D Basic Equations from Fluid Mechanics	674	(17)
D.1 Introduction	674	(1)
D.2 Definitions and Concepts	675	(2)
D.3 Laws of Motion	677	(6)
D.3.1 Differential Continuity Equation	678	(1)
D.3.2 Differential Momentum Equation (Navier-Stokes Equations)	678	(2)
D.3.3 Thermal Energy Equation	680	(1)
D.3.4 Conservative Form of Equations	680	(2)
D.3.5 Supplementary Equations	682	(1)
D.3.6 Problem Statement	682	(1)
D.4 Stream Functions and Vorticity	683	(2)
D.5 Potential Flow	685	(1)
D.6 Viscous Incompressible Flow	686	(3)
D.6.1 Primitive Variable Formulation	686	(1)
D.6.2 Vorticity and Stream Function Formulation	687	(2)
D.7 Boundary Layer Flow	689	(2)
References	690	(1)
Appendix E Basic Equations from Heat Transfer	691	(18)
E.1 Introduction	691	(1)
E.2 Conduction	692	(4)
E.2.1 Heat Conduction Equation	693	(1)
E.2.2 Boundary Conditions	694	(2)
E.2.3 Nondimensional Parameters	696	(1)
E.3 Convection	696	(5)
E.3.1 Convection Equations	697	(2)
E.3.2 Boundary Conditions	699	(1)
E.3.3 Nondimensional Parameters	700	(1)
E.4 Radiation	701	(6)
E.4.1 Surface Radiation	702	(2)
E.4.2 Radiation Exchange between Surfaces	704	(3)
E.5 Heat Transfer Units	707	(2)
References	708	(1)
Index	709